Explicit formulas for pricing of callable mortgage-backed securities in a case of prepayment rate negatively correlated with interest rates

In this paper, we deal with the pricing of Mortgage-Backed Securities (MBS) in the reduced-form framework. Based on the ideas presented by Brunel and Jribi (2008) [8] and Rom-Poulsen (2007) [7], we introduce a stochastic process Qt=e−∫0tλsds to model the prepayment factor and assume that the prepayment rate λtλt is inversely proportional to the stochastic interest rate rtrt, which follows a CIR process. Explicit formulas for pass-through MBSs and semi-analytical solutions for Collateralized Mortgage Obligations (CMO) are obtained through PDE approaches. Based on the formulas, numerical results are provided to explain the dependence of MBS prices on mortgage parameters and the negative correlation between MBS prices and interest rates.